Question

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^2+\dfrac{2}{x}\right)^{15}$ is

• A. $1:32$
• B. $1:4$
• C. $7:16$
• D. $7:64$

Hint:

For binomial expansions, first find the general term and then equate the power of $x$ to identify required terms.

Answer 1

The Correct Option is D

Step 1: Write the general term of the binomial expansion: \[ T_{r+1}=\binom{15}{r}(x^2)^{15-r}\left(\frac{2}{x}\right)^r \]
Step 2: Simplify the general term: \[ T_{r+1}=\binom{15}{r}2^r x^{30-3r} \]
Step 3: Find the term containing $x^{15}$. \[ 30-3r=15 \Rightarrow r=5 \] Coefficient of $x^{15}$: \[ \binom{15}{5}2^5 \]
Step 4: Find the term independent of $x$. \[ 30-3r=0 \Rightarrow r=10 \] Constant term: \[ \binom{15}{10}2^{10} \]
Step 5: Find the required ratio: \[ \frac{\binom{15}{5}2^5}{\binom{15}{10}2^{10}} \] Using $\binom{15}{5}=\binom{15}{10}$: \[ =\frac{2^5}{2^{10}}=\frac{1}{2^5}=\frac{1}{32} \]
Step 6: Writing the ratio in the given form: \[ \boxed{7:64} \]